Complex Number

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Basic Concepts

Definition: $i=\sqrt{-1}$

Complex number: $a+bi$, where $a$ is real part，$bi$ is imaginary part. $a={\rm Re}(\mathbb{Z})$, $b={\rm Im}(\mathbb{Z})$

Notes: $i^2=-1$, $i^3=-i$, $i^4=1$

It can also be written in vector form, such as $a \choose b$ (Note: this writing is only useful when calculating addition and subtraction, other operations are more troublesome).

Calculation:

Addition/Subtraction: $(a+bi)\pm (c+di)=(a+c)\pm(b+d)i$

Multiplication: $(a+bi)(c+di)=(ac-bd)+(bc+ad)i$

Division: similar to rationalizing the denominator, i.e.
$$\frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{(c+di)(c-di)}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}$$

Modulus) $|z|=\sqrt{a^2+b^2}$. 其中: $$|z_1z_2|=|z_1||z_2|, \quad
|\frac{z_1}{z_2}|=\frac{|z_1|}{|z_2|}$$

Complex conjugate: For the complex number $z=a+bi$, the conjugate is $\bar{z}=a-bi$. These two complex numbers are conjugate to each other and have the following properties:
$$ z \cdot \bar{z} = a^2+b^2 = |z|^2$$

Argument: in argand plane, the angle measured from the positive real axis to the line segment of the complex number, $${\rm arg}(z_1z_2)={\rm arg}(z_1){\rm arg}(z_2), \quad {\rm arg}(\frac{z_1}{z_2})={\rm arg}(z_1)-{\rm arg}(z_2)$$

Other expressions for complex number:

• Polar form: $z=r \cos\theta + i r\sin\theta$, where $\theta$ is argument, r is modulus of it.
• Using Euler Formula, it can be changed into modulus-argument form: $z=re^{i\theta}$. This form is more convenient in calculating the multiplication and division of complex numbers, that is, $z_1 \cdot z_2 = r_1 r_2 e^{i(\theta_1+\theta_2)}$.

Loci

• $|z|=r$: a circle radius r centre the origin.
• $|z-c|=r$: a circle radius r centre c.
• $|z−a| = |z−b|$: z is equidistant from both a and b, i.e. the locus is the perpendicular bisector of the line connecting a and b.
• ${\rm arg}(z) = \theta$: the half line from the origin making angle θ from the positive x-axis (measured anti-clockwise).
• ${\rm arg}(z-c) = \theta$: the half line from point c making angle θ with the horizontal.
• ${\rm arg}(\frac{z-a}{z-b})=\alpha$: The circle that passes through a and b and whose angle of the arc ${ab}$is $\alpha$.

Transformation

• $z\mapsto z+a$: a translation by complex number a (which can be thought of as a vector).
• $z\mapsto \lambda z$: an enlargement scale factor $\lambda$ centre the origin, where $\lambda$ is real.
• $z \mapsto z(\cos \theta + i\sin\theta)$: rotation about the origin by angle θ anticlockwis.